ABSTRACT
In Chapter 3, we considered variants of Sperner’s theorem, and in Chapter 4 we saw random versions of Theorem 1. In this chapter we will consider an area of extremal set theory that can be described by the poset structure of 2[n]. Any family F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429440809/2714d8f8-ef71-44f4-80bb-22c5d5471b0d/content/eq6397.tif"/> ⊆ 2[n] of sets is a poset under the inclusion relation, so any property described in the language of ordered sets can be translated to a property of set families. We say that a subposet Q′ of Q is a (weak) copy of P, if there exists a bijection ƒ : P → Q′, such that for any p, p′ ∈ P, the relation p < P p′ implies f (p) < Q f (p′). We say that Q′ is a strong or induced copy of Q, if, in addition to this, f (p) < Q f (p′) also implies p < P p′. If a poset Q does not contain a weak copy of P, then it is P-free. Katona and Tarján [344] introduced the problem of determining L a ( n , P ) = max { | F | : F ⊆ 2 [ n ] is P -free} . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429440809/2714d8f8-ef71-44f4-80bb-22c5d5471b0d/content/eq6398.tif"/>
